In this article, we present the complete conservative N\(^3\) LO spin-orbit interaction,
using the framework of EFT [1]}, [2]} and an extension of the diagrammatic approach presented in [3]}. In particular,
we begin with deriving the required Feynman rules and then the Feynman diagrams:
\(1 \, (0\text{PN}) + 4 \, (1\text{PN}) + 21 \, (2\text{PN}) + 130 \, (3\text{PN})\)  = 156, for the non-spinning sector;
\( 2 \, (\text{LO}) + 13 \, (\text{NLO}) + 100 \, (\text{N$^2$LO }) + 894 \, (\text{N$^3$LO }) \)  = 1009, for the spinning sector,
The corresponding tensor integrals are decomposed in terms of scalar loop integrals, up to three loops, which are decomposed in terms of a minimal basis of master integrals (MIs), by means of integration-by-parts (IBP) identities [4]}, [5]}. The contribution of each diagram to the scattering amplitude is then obtained after substituting the analytic expression of the MIs. Finally,
the Fourier transform of the amplitude generates the the effective Lagrangian, later converted into a Hamiltonian which is the main result of this article.
The computational setup has been fully automated within an in-house
Mathematica routine
interfaced to
QGRAF [6]}, used for the diagram generation,
xTensor [7]}, for tensor algebra manipulation,
LiteRed [8]}, for the IBP decomposition,
elaborating on some of the ideas implemented in EFTofPNG [9]}.
